Given an arbitrary group G, there is a related profinite group G < sup >^</ sup >, the profinite completion of G. It is defined as the inverse limit of the groups G / N, where N runs through the normal subgroups in G of finite index ( these normal subgroups are partially ordered by inclusion, which translates into an inverse system of natural homomorphisms between the quotients ).

Given a probability space, a filtration is a weakly increasing collection of sigma-algebras on,, indexed by some totally ordered set T, and bounded above by.

Given the extremely disadvantaged position in which the English were, Vernon ordered a retreat but not before throwing some 300 bombs on the city.

Given the distance that these troops had to march in order to make contact with the enemy, it was clear to the two Corps commanders that they would be unable to attack at 4: 00 as ordered.

Given intelligence that the rebels had been stockpiling weapons at Concord, Massachusetts, he ordered a troop of British regulars to march there on the night of 18 April to confiscate them.

Given a non-empty totally ordered set R with the following four properties:

Given an orthonormal basis ordered such that, we see that

Given a partially ordered class P, we can form a category with P's elements as objects and a single arrow going from one object ( element ) to another if the first is less than or equal to the second.

Given 25, 000 men, he was ordered to clear Confederate obstacles on the Mississippi River.

Given two partially ordered sets A and B, the lexicographical order on the Cartesian product A × B is defined as

Given points a, b, c, and d on a line, if it is known that the points are ordered as ( a, b, c ) and ( b, c, d ), then it is also true that ( a, b, d ).

#: Given three ordered periods,,, the price index for periods and times the price index for periods and should be equivalent to the price index for periods and.

Given that the documentary had aroused much tension among the Party ’ s top leaders, the Ministry of Radio, Film and Television ordered CCTV to revise the original version of the film before it could be broadcast again.

Given an ordered choice of linearly independent eigenvectors, especially an eigenbasis, they can be indexed by eigenvalues, i. e. using a double index, with x < sub > i, j </ sub > being the j < sup > th </ sup > eigenvector for the i < sup > th </ sup > eigenvalue.

Given its inner-city roots, chicken and food is quite often ordered a la carte.

Given an arithmetic function, one can generate a bi-infinite sequence of other arithmetic functions by repeatedly applying the first summation.

: Given any positive number ε, there is a sequence

# Given any point x in X, and any sequence in X converging to x, the composition of f with this sequence converges to f ( x )

Given a class function G: V → V, there exists a unique transfinite sequence F: Ord → V ( where Ord is the class of all ordinals ) such that

Given any two similar polygons, corresponding sides taken in the same sequence are proportional and corresponding angles taken in the same sequence are equal in measure.

* Given an infinite sequence of infinite strings, where each character of each string is chosen uniformly at random, any given finite string almost surely occurs as a prefix of one of these strings.

Solomonoff's universal prior probability of any prefix p of a computable sequence x is the sum of the probabilities of all programs ( for a universal computer ) that compute something starting with p. Given some p and any computable but unknown probability distribution from which x is sampled, the universal prior and Bayes ' theorem can be used to predict the yet unseen parts of x in optimal fashion.

Given a bounded sequence, there exists a closed ball that contains the image of ( is a subset of the scalar field ).

Given a testing procedure E applied to each prepared system, we obtain a sequence of values

Given a base for a topology, in order to prove convergence of a net or sequence it is sufficient to prove that it is eventually in every set in the base which contains the putative limit.

Given this, it is quite natural and convenient to designate a general sequence a < sub > n </ sub > by by the formal expression, even though the latter is not an expression formed by the operations of addition and multiplication defined above ( from which only finite sums can be constructed ).

Given a short exact sequence with maps q and r:

Given a linearly recursive sequence, let C be the transpose of the companion matrix of its characteristic polynomial, that is

Given this hypothesis that a novel FOXP2 sequence can aid echolocation, echolocating and non echolocating cetaceans might be predicted to display differences in their FOXP2 sequences.

Given a strictly increasing integer sequence / function ( n ≥ 1 ) we can produce a faster growing sequence ( where the superscript n denotes the n < sup > th </ sup > functional power ).

Given a sequence of positive integers, the Gödel encoding of the sequence is the product of the first n primes raised to their corresponding values in the sequence:

Given the observation space, the state space, a sequence of observations, transition matrix of size such that stores the transition probability of transiting from state to state, emission matrix of size such that stores the probability of observing from state, an array of initial probabilities of size such that stores the probability that. We say a path is a sequence of states that generate the observations.

Given two sequences X and Y, a sequence G is said to be a common subsequence of X and Y, if G is a subsequence of both X and Y.

* Given any topological space X, the continuous real-or complex-valued functions on X form a real or complex unitary associative algebra ; here the functions are added and multiplied pointwise.

Given a finite dimensional real quadratic space with quadratic form, the geometric algebra for this quadratic space is the Clifford algebra Cℓ ( V, Q ).

Given a vector space V over the field R of real numbers, a function is called sublinear if

Given a function f of a real variable x and an interval of the real line, the definite integral

Given the tendency for real negotiations in Japan to be conducted privately ( in the nemawashi, or root binding, process of consensus building ), the shingikai often represented a fairly advanced stage in policy formulation in which relatively minor differences could be thrashed out and the resulting decisions couched in language acceptable to all.

Given m real values t < sub > i </ sub >, called knots, with

Given metric spaces ( X, d < sub > 1 </ sub >) and ( Y, d < sub > 2 </ sub >), a function f: X → Y is called uniformly continuous if for every real number ε > 0 there exists δ > 0 such that for every x, y ∈ X with d < sub > 1 </ sub >( x, y ) < δ, we have that d < sub > 2 </ sub >( f ( x ), f ( y )) < ε.

Given two metric spaces ( X, d < sub > X </ sub >) and ( Y, d < sub > Y </ sub >), where d < sub > X </ sub > denotes the metric on the set X and d < sub > Y </ sub > is the metric on set Y ( for example, Y might be the set of real numbers R with the metric d < sub > Y </ sub >( x, y )

: Given: a function f: A R from some set A to the real numbers

Given expectations about returns on fixed investment, every level of the real interest rate ( i ) will generate a certain level of planned fixed investment and other interest-sensitive spending: lower interest rates encourage higher fixed investment and the like.

Given two C < sup > k </ sup >- vector fields V, W defined on S and a real valued C < sup > k </ sup >- function f defined on S, the two operations scalar multiplication and vector addition

Given real life prisons are same-sex communities, this fetish does lend itself to male on male or female on female activities and settings.

Given that the Liber historiae Francorum admits four years for Chlothar's reign, it remains possible that Chlothar began to exert some real authority when his mentors deemed him an adult, 669.

Given that his predecessors were monarchists who tried without success to restore the French monarchy, Grévy is seen as the first real republican President of France.

Given the variety of approaches, Economic Geography has taken to many different subject matters, including: the location of industries, economies of agglomeration ( also known as " linkages "), transportation, international trade, economic development, real estate, gentrification, ethnic economies, gendered economies, core-periphery theory, the economics of urban form, the relationship between the environment and the economy ( tying into a long history of geographers studying culture-environment interaction ), and globalization.

Given the nature of this interrogation and the motive behind it, it is not known for certain what elements of Number Six's life so portrayed are real and which are fiction.

Given two normed vector spaces V and W ( over the same base field, either the real numbers R or the complex numbers C ), a linear map A: V → W is continuous if and only if there exists a real number c such that

Given a real dynamical system ( R, M, Φ ), I ( x ) is an open interval in the real numbers, that is.

* Given a positive real number ε, an ε-isometry or almost isometry ( also called a Hausdorff approximation ) is a map between metric spaces such that

Given an m × n matrix with real entries ( or entries from any other field ) and rank r, then there exists at least one non-zero r × r minor, while all larger minors are zero.

Given that bankruptcies and real estate prices did not fare as negatively in Central Canada as in the rest of Canada and the United States during the NEP, it is possible that the NEP had a positive effect in Central Canada.